an equivalence relation
An equivalence relation is the precise way to say 'the same in the way we care about', even when the things are not literally identical. Two coins of the same value, two words that rhyme, two people born in the same year — each is a 'same as' judgement that ignores everything except the feature you chose. The point is to sort a big messy collection into clean groups where everything inside a group counts as interchangeable.
A relation R is an equivalence relation when it has all three of these properties at once: reflexive (everything is equivalent to itself, x R x), symmetric (if x R y then y R x), and transitive (if x R y and y R z then x R z). The everyday relation 'has the same remainder when divided by 3' on the whole numbers is a perfect example: 4 and 7 are equivalent (both leave remainder 1), the relation obviously holds between a number and itself, it does not care which number you name first, and remainders chain together transitively.
This single idea is the engine behind two cornerstone results. The Myhill-Nerode theorem declares two strings equivalent when no continuation can ever tell them apart for a given language; the number of resulting groups tells you exactly how many states the smallest DFA needs, and is finite if and only if the language is regular. DFA minimization works by merging states that are equivalent in the sense that they accept exactly the same future strings. In both cases an equivalence relation slices an infinite set of strings or states into a tidy collection of classes.
On the integers, 'a and b leave the same remainder mod 3' is an equivalence relation. It splits every integer into three groups: remainder 0 ({..., -3, 0, 3, 6, ...}), remainder 1, and remainder 2. Any two members of the same group are interchangeable for this purpose.
Same-remainder-mod-3 is reflexive, symmetric, and transitive, so it sorts the integers into 3 classes.
All three properties are required. Drop symmetry and 'less than or equal' qualifies but is not an equivalence; drop transitivity and 'differs by at most 1' fails. Missing even one property breaks the partition.